A Theory of Diversity for Random Matrices with Applications to In-Context Learning of Schr\"odinger Equations

TL;DR

This paper develops a probabilistic framework for understanding the triviality of centralizers in collections of random matrices, with applications to the generalization of transformer neural networks in quantum physics modeling.

Contribution

It introduces bounds on the probability of trivial centralizers for random matrices from Schr"odinger discretizations, linking random matrix theory to machine learning guarantees.

Findings

Lower bounds on trivial centralizer probabilities for specific matrix families.

Application of results to in-context learning of Schr"odinger equations.

Implications for neural network generalization in quantum physics tasks.

Abstract

We address the following question: given a collection $\{\mathbf{A}^{(1)}, \dots, \mathbf{A}^{(N)}\}$ of independent $d \times d$ random matrices drawn from a common distribution $\mathbb{P}$, what is the probability that the centralizer of $\{\mathbf{A}^{(1)}, \dots, \mathbf{A}^{(N)}\}$ is trivial? We provide lower bounds on this probability in terms of the sample size $N$ and the dimension $d$ for several families of random matrices which arise from the discretization of linear Schr\"odinger operators with random potentials. When combined with recent work on machine learning theory, our results provide guarantees on the generalization ability of transformer-based neural networks for in-context learning of Schr\"odinger equations.